J. ALGEBRAIC GEOMETRY
00 (XXXX) 000–000
S 1056-3911(XX)0000-0

IWASAWA THEORY OF THE FINE SELMERGROUPCHRISTIAN WUTHRICH

Abstract
The fine Selmer group of an elliptic curve E over a number field K is
obtained as a subgroup of the usual Selmer group by imposing stronger
conditions at places above p. We prove a formula for the Euler-characteristic of the fine Selmer group over a Zp -extension and use it tocompute explicit examples.

1. Introduction
Let E be an elliptic curve defined over a number field K and let p be an
odd prime. We choose a finite set of places Σ in K containing all places above
p · ∞ and such that E has good reduction outside Σ. The Galois group of the
maximal extension of K which is unramified outside Σ is denoted by GΣ (K).
In everything that follows ⊕Σ always stands for the product over all finite
places υ in Σ. Let E{p} be the GΣ (K)-module of all torsion points on E
whose order is a power of p. For a finite extension L : K, the fine Selmer
group is defined to be the kernel0 R(E/L) H 1 (GΣ (L), E{p}) ⊕Σ H 1 (Lυ , E{p})where H i (Lυ , ·) is a shorthand for the product ⊕w|υ H i (Lw , ·) over all places
w in L above υ. If L is an infinite extension, we define R(E/L) to be theinductive limit of R(E/L0 ) for all finite subextensions L : L0 : K. Note that
R(E/L) does not depend on the choice of the set Σ.
The fine Selmer group has often appeared in the Iwasawa theory of elliptic
curves and has different names such as the “strict” or “restricted” Selmer
group. We stick to the terminology in [5].
In this article we will be concerned with the behaviour of the fine Selmer
group in a given Zp -extension ∞ K : K which will often be the cyclotomic
Zp -extension. Let Γ be the Galois group of ∞ K : K and let Λ be the Iwasawaalgebra of Γ. Given a topological generator γ of Γ, we may identify Λ with
Received November 8, 2005.
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CHRISTIAN WUTHRICH

Zp [[T ]]. It is well known that the Pontryagin-dual of R(E/∞ K) is a finitely
generated Λ-module.
The same is true of the dual of the classical p-primary Selmer group
S(E/∞ K). We recall the important results of Perrin-Riou [10] and Schneider [16]. If p is assumed to be an odd prime such that E has good ordinary
reduction at all places above p, then there exists a canonical p-adic heightpairing on the Selmer group whose regulator is linked to the Iwasawa theoryof the Selmer group S(E/∞ K). More precisely, if the p-primary part of theTate-Shafarevich group X(E/K){p} is finite and if the regulator does notvanish then the dual of S(E/∞ K) is a torsion Λ-module and the order ofvanishing of its characteristic power series fS ∈ Λ at T = 0 is equal to thecorank of S(E/K). Moreover the leading coefficient fS∗ (0) of fS at T = 0 can
be computed asQcυ · Np2 · #X(E/K){p}∗(mod Z×(1.1)
fS (0) ≡ Regp (E/K) · υp)(#E(K){p})2where Regp (E/K) is the normalised p-adic regulator on E(K), Np is the
product of the number of points on the reduction at the places above p and cυ
is the Tamagawa number of E at υ. Note the importance of the assumptionthat E has ordinary reduction above p. See for instance [8] for results in the
supersingular case.
This article proves the analogous results for the fine Selmer group. One of
the important differences is that we are allowed to drop all conditions on theodd prime p. In particular E may have bad reduction of any type and thesupersingular situation does not seem to differ in any way from the ordinary
situation.
The compact version of the fine Selmer group R(E/K) is defined to be thekernel of the following localisation map
(1.2)

0

 R(E/K)

 H 1 (GΣ (K), Tp E) ⊕υ|p H 1 (Kυ , Tp E)

where Tp E = lim E[pk ] is the Tate-module of E. It will be shown in Lemma 3.1←−
that R(E/K) is a free Zp -module whose rank is equal to the corank ofR(E/K).
Perrin-Riou has defined in [11] and [12] a p-adic height pairing on the fineSelmer group R(E/K) affiliated with the chosen Zp -extension. We recall
the definition in section 5. Perrin-Riou made the conjecture that the heightpairing is non-degenerate. The regulator of the p-adic hieght may be linked
to the Iwasawa theory of the fine Selmer group, see Proposition 5.3. It willappear in the formula of the leading term of the characteristic series fR of thedual of the fine Selmer group R(E/∞ K). Other arithmetic invariants of E

IWASAWA THEORY OF THE FINE SELMER GROUP

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also appear, including the order of the fine Tate-Shafarevich group (E/K)
with respect to p as defined in [19]. See section 6 for a definition.
We state here the main theorem only in the special situation when theelliptic curve is defined over Q; see Theorem 6.1 for the general statement. If
E is an elliptic curve over Q a much celebrated theorem of Kato [7, Theorem
12.4] states that the dual of R(E/∞ Q) is a torsion Λ-module (the so called
weak Leopoldt conjecture for E). In the general case this can be checked via
the non-degeneracy of the p-adic height on the fine Selmer group. Similarly
it is known that the order of vanishing of fR at T = 0 is equal to the corankof R(E/K), if the regulator is non-zero; see [11, Proposition 3.4.5] and [12,Corollaire 3.4.3].
Let ∞ Q be the cyclotomic Zp -extension of Q and let n Q be its nth layer.
Theorem 1.1. Let E/Q be an elliptic curve whose Tate-Shafarevich group
X(E/Q) has finite p-primary part. Suppose that the p-adic height on thefine Selmer group is non-degenerate. Then there is an injection with finite
cokernel J of R(E/Q) into the cokernel of the corestriction map
cor: lim H 1 (GΣ (n Q), Tp E)←−

 H 1 (GΣ (Q), Tp E).

n

If the reduction is potentially good, then the leading coefficient of the characteristic power series of the dual of the fine Selmer group R(E/∞ Q) is equaltoQcυ · # TorsZp (D) · #(E/Q)∗(mod Z×(1.3) fR (0) ≡ Reg(R(E/Q)) · υp)#Jwhere D is the cokernel of the localisation map from E(Q) ⊗ Zp to the p-adic
completion of E(Qp ).
The proof of the theorem is given in section 7. See Corollary 6.3 for thestatement split up according to whether E has rank 0, 1 or greater than 1.
In sections 9, 10 and 11 we include a list of interesting examples. Unlike
for the formula 1.1, the expression on the right hand side is not explicitly
computable due to the presence of the unknown size of the cokernel J. Nevertheless the formula is still very useful to determine fR or even R(E/∞ Q).
We show how to use the formulae (1.3) and (1.1) and analytic informationfrom [13].
For almost all cases, the result is that fR is a simple power of T ; in otherwords that R(E/∞ Q) has the same corank as R(E/Q) and that (E/∞ Q)
is still finite, if (E/Q) is finite. This provides ample evidence that the µinvariant of the fine Selmer group should always be trivial as conjectured by
Coates and Sujatha in [5, Conjecture A].

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CHRISTIAN WUTHRICH

It is not true that fR is always a power of T . The discussion of an example
in which the corank of the fine Selmer group over ∞ Q is strictly larger thanover Q is included in section 11. The proof relies on the computation of therank of the Mordell-Weil over the first two layers of the Z3 -extension.
So far, we cannot find a single example of a curve E/Q and an odd prime
p for which we can prove that the fine Tate-Shafarevich group (E/∞ Q) is
infinite. See also Question 8.3 in [19].
Motivated by Proposition 9.2, we make the following
Conjecture 1.2. Let E/Q be an elliptic curve. The set of primes p forwhich the corank of R(E/∞ Q) is larger than the corank of R(E/Q) is finite.
It is a pleasure to thank John Coates, Robert Pollack and Karl Rubin for
helpful discussions.

2. Notations
If f is a map between Zp -modules with finite kernel and cokernel, we writez(f ) =# ker(f ).
# coker(f )For any p-primary abelian group A, the expression Adiv stands for the
maximal divisible subgroup of A and A/ div for the quotient of A by Adiv .
The Pontryagin dual of an abelian group A is written Â and its p-primarypart is denoted by A{p}.
To ease notations of Galois-cohomology we will use the following shorter
notations for any GΣ (K)-module M :HΣi (K, M ) = H i (GΣ (K), M )HΣi (∞ K, M ) = H i (GΣ (∞ K), M )The following projective limits along the corestriction maps will often appearii∞ H (E/K) = lim H (GΣ (n K), Tp E)←−nwhere Tp E is the Tate module lim E[pk ]. The Γ-module ∞ H i (E/K) does not←−depend on the choice of the finite set Σ containing all bad places, all places
above p and above ∞. It is trivial for i 6= 1, 2.
If υ is a place in K and L is an extension of K, the following notation will
be usedH i (Lυ , M ) = ⊕w|υ H i (Lw , M )whenever M is a module under the absolute Galois group of Kυ .

IWASAWA THEORY OF THE FINE SELMER GROUP

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The fine Selmer group R(E/K) was already defined in the beginning of theintroduction. There are also two compact versions of the fine Selmer group;
the first is defined by the exact sequence (1.2) and the second smaller groupis RΣ (E/K), obtained by imposing the local conditions at all places in Σ:0 RΣ (E/K)

 HΣ1 (K, Tp E) ⊕Σ H 1 (Kυ , Tp E)

It is not difficult to see that RΣ (E/K) has finite index in R(E/K), sinceH 1 (Kυ , Tp E) is finite if υ does not divide p.
A Zp -extension ∞ K : K can be given as the fixed field of the kernel of ahomomorphism λ : GΣ (K)  Zp . Conversely a surjective morphism λ in
Hom(GΣ (K), Zp ) is determined by ∞ K up to multiplication by a unit in Z×p.
In particular, if σ is an element in GΣ (K), then its action on a basis ζ ∈ Tp µ1
is given by σ(ζ) = ζ χ(σ) for some χ(σ) in Z×
p . Then λ = p · logp ◦χ defines thecyclotomic Zp -extension cyc K of K, where logp : Z×
 pZp is the p-adicplogarithm.
2.1. Global duality. For the proof of the main theorem, the global duality of Poitou-Tate (see [9, Theorem 8.6.8]) will be used. Over the field K,
we can extract a five term exact sequence0 R(E/K) HΣ1 (K, E{p}) ⊕Σ H 1 (Kυ , E{p})RΣ (E/K)∧ ≺0≺HΣ1 (K, Tp E)∧ ≺and a four term exact sequence0 E(K){p} ⊕Σ E(Kυ ){p}

 HΣ2 (K, Tp E)∧

 R(E/K)

 0.

Finally, it also shows that RΣ (E/K) is dual to the kernel of the localisation
map from HΣ2 (K, E{p}) to ⊕Σ H 2 (Kυ , E{p}). Since the target of this map istrivial by local Tate duality, RΣ (E/K) is dual to HΣ2 (K, E{p}).
The global duality over ∞ K can be formulated similarly, e.g. the five term
sequence becomes:0 R(E/∞ K) HΣ1 (∞ K, E{p})⊕ H 1 ( K , E{p})
Σ
∞ υ

0≺

HΣ2 (∞ K, E{p}) ≺∞H1(E/K)∧ ≺6

CHRISTIAN WUTHRICH

3. The compact fine Selmer group is a free Zp -module
The snake lemma can be applied to the following commutative diagramwith exact rows.
0 E(K){p} HΣ1 (K, Tp E)
 Tp HΣ1 (K, E{p})00g ⊕Σ E(Kυ ){p}g
 ⊕Σ H 1 (Kυ , Tp E)g ⊕Σ Tp H 1 (Kυ , E{p})  0

Using that the functor Tp is left-exact, we see that there is an exact sequence0 RΣ (E/K)

 Tp R(E/K)

 TΣ

where TΣ is the finite cokernel of the vertical map on the left hand side. The
sums in the bottom exact sequence of the previous diagram can be replacedby the sums running only over the places above p, one obtains an injection ofR(E/K) into Tp R(E/K) of finite index. Hence we have proved that
Lemma 3.1. The compact fine Selmer groups R(E/K) and RΣ (E/K) are
free Zp -modules contained in Tp R(E/K).
A counter-example showing that R(E/K) is not always equal to Tp R(E/K)
is given in (9) of [19].
In particular the lemma provides us with a map a defined as the dual of
the composition
 (R(E/K)div )∧
w
w
w
w
w

R(E/K)∧(3.1)

HomZp (Tp R(E/K), Zp )  HomZp (RΣ (E/K), Zp )∧Hence a is a map from HomZp (RΣ (E/K), Zp ) to R(E/K) whose kernel hasorder# ker(a) = [Tp R(E/K) : RΣ (E/K)]= [Tp R(E/K) : R(E/K)] · [R(E/K) : RΣ (E/K)]
and the cokernel is dual to R(E/K)/ div. In other words, we have
(3.2)

z(a) =[Tp R(E/K) : R(E/K)] · [R(E/K) : RΣ (E/K)].
#(R(E/K)/ div)

4. Control theorem
We repeat here for the sake of completeness the proof of the well-known
control theorem for the fine Selmer group. See for instance Proposition 7.4.4
in [15].

IWASAWA THEORY OF THE FINE SELMER GROUP

7

Proposition 4.1. The restriction map b : R(E/K)
 R(E/∞ K)Γ has
finite kernel and cokernel. The kernel has no more elements than E(K){p}Q
Q
and the cokernel has no more elements than υ|p E(Kυ ){p} · υ-p cυ , wherecυ is the Tamagawa number of E at υ.
Proof. We consider the following commutative diagram with exact rows

(4.1)

 R(E/∞ K)Γ
fb

 HΣ1 (∞ K, E{p})Γ
ff

 ⊕Σ H 1 (∞ Kυ , E{p})Γ
ff0

 R(E/K) HΣ1 (K, E{p}) ⊕Σ H 1 (Kυ , E{p})0

The Hochschild-Serre spectral sequence yields that the middle vertical map issurjective and its kernel is the group Tgl = H 1 (Γ, E(∞ K){p}); similarly the
kernel of the vertical map on the right is equal to
Tloc = ⊕Σ H 1 (Γ, H 0 (∞ Kυ , E{p})).
So there are two exact sequences deduced from the spectral sequence of
Hochschild-Serre
0
(4.2)
0

 Tgl

 HΣ1 (K, E{p}) HΣ1 (∞ K, E{p})Γ

0

 Tloc ⊕Σ H 1 (Kυ , E{p}) ⊕Σ H 1 (∞ Kυ , E{p})Γ

0

The next two lemmata will finish the proof of the proposition.

Lemma 4.2. The group Tgl is a finite group of order bounded by E(K){p}.
Moreover, if ∞ K is the cyclotomic Zp -extension and E has potentially good
reduction at all primes above p, then Tgl has exactly as many elements as
E(K){p}.
Proof. Let M be the Γ-module E(∞ K){p} and so Tgl = H 1 (Γ, M ). If
M is finite, and this is the case under the more restrictive hypothesis of thesecond statement (see [6]), then #H 1 (Γ, M ) = #H 0 (Γ, M ) and so #Tgl =#E(K){p}. Now let D be the maximal divisible subgroup of M and consider
the exact sequence of Γ-modules 0 Tp D Vp DD 0, where
Tp D is lim D[pk ] and Vp D = Tp D ⊗ Qp . Since H 1 (Γ, Vp D) and H 2 (Γ, Tp D)←−
vanish, we see that D Γ is finite and H 1 (Γ, D) is trivial. Now the cohomology
D
M
 M/D
 0 showsof the short exact sequence 0
that Tgl = H 1 (Γ, M ) is equal to H 1 (Γ, M/D) which has the same numberof elements as H 0 (Γ, M/D). This latter group is equal to the quotient of
M Γ = E(K){p} by D Γ , because H 1 (Γ, D) vanishes. Hence#Tgl =#E(K){p}
.
#DΓ


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CHRISTIAN WUTHRICH

Lemma 4.3. Let υ be a finite place of K. The groupTυ = H 1 (Γ, H 0 (∞ Kυ , E{p})is a finite group of order bounded by the order of E(Kυ ){p}. If υ does not lie
above p, its order is equal to the highest power of p dividing the Tamagawa
number cυ . If ∞ K is the cyclotomic Zp -extension, υ divides p and the reduction of E at υ is potentially good, then the order of Tυ is equal to the order ofE(Kυ ){p}.
Proof. Let w be any place of ∞ K above υ. By Shapiro’s lemma, Tυ isisomorphic to H 1 (Γw , E(∞ Kw ){p}) where Γw is the decomposition group of
Γ at w. If υ splits completely in ∞ K, then Γw is trivial and the statement isclear. Otherwise Γw is isomorphic to Zp and the proof of the bound can becopied from the proof of the previous lemma.
Finally the statement for the case that υ does not lie above p is containedin Lemma 3.4 of [4].


5. The height pairing
In this section, we construct the p-adic height pairing on the fine Selmer
group. The construction follows closely the original definition of Perrin-Riouin [11], but its presentation is simplified and adapted to our needs. We makea further assumption1 on the finite set Σ, namely we impose that the classgroup of the Σ-integers in K is trivial. This can be achieved by adding a setof generators of the class group to the set Σ. The final result in Theorem 6.1
is independent of this choice.
 ξσ be a 1-cocycle representing an el5.1. Extensions. Let ξ : σement of the compact fine Selmer group RΣ (E/K). Its class belongs to
H 1 (GΣ (K), Tp E), which is equal to Ext1GΣ (K) (Tp E, Zp (1)), i.e. it corresponds
to a short exact sequence
(5.1)

0

 Zp (1) Tξ

 Tp E0of GΣ (K)-modules. Explicitly this can be constructed in the following manner:
As a Zp -module Tξ is just the direct sum Zp (1) ⊕ Tp E, but with the twistedGΣ (K)-action given by the the formula(ζ, Q)σ = (ζ σ + hξσ , Qσ i, Qσ ) for all ζ ∈ Zp (1), Q ∈ Tp E and σ ∈ GΣ (K),
where we denoted by h·, ·i the Weil-pairing Tp E ⊗ Tp E
1This is not strictly necessary but simplifies the exposition.

 Zp (1).

IWASAWA THEORY OF THE FINE SELMER GROUP

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If υ is a place in Σ and Gυ any decomposition group at υ, then the sequence (5.1) is a split exact sequence of Gυ -module. This follows directlyfrom the definition of RΣ (E/K), given that resυ (ξ) is trivial in H 1 (Kυ , Tp E).
Lemma 5.1. We have the following diagram0 HΣ1 (K, Zp (1))

 HΣ1 (K, Tξ )

 HΣ1 (K, Tp E)

0

0

g
 ⊕Σ H 1 (Kυ , Zp (1))

g ⊕Σ H 1 (Kυ , Tξ )g ⊕Σ H 1 (Kυ , Tp E)

0Proof. It is already clear why the bottom line is exact. The triviality inthe upper left hand corner is due to the fact that Tp E(K) = 0. On the righthand side we should complete the diagram with the following squareHΣ1 (K, Tp E)

 HΣ2 (K, Zp (1))

g⊕Σ H (Kυ Tp E)g ⊕Σ H (Kυ , Zp (1))

1

2

But we know that the bottom map is trivial and, by global class field theory(see [9, Theorem 8.6.3]), that the map on the right hand side is injective byassumption of the triviality of the class group of the Σ-integers. This proves
the lemma.

We will now apply the snake lemma to the diagram in the previous lemma.
The kernel of the vertical map on the right hand side is precisely RΣ (E/K).
For the left hand side, the global duality of Poitou-Tate, see [9, Theorem
10.3.12], gives an exact sequenceHΣ1 (K, Zp (1))

 ⊕Σ H 1 (Kυ , Zp (1))

 HΣ1 (K, Qp/Zp )∧0in which the zero at the end is a consequence of the assumption on the classgroup of the Σ-integers. The last non-zero term is actually the Galois group
GΣ (K)p-ab of the maximal abelian p-extension of K which is unramified outside Σ. The snake lemma provides us with a map
Hξ : RΣ (E/K)

 GΣ (K)p-ab

and hence for every Zp -extension given by a morphism λ : GΣ (K)obtain a pairingh·, ·iλ : RΣ (E/K) × RΣ (E/K)(ξ, η)

 Zp we Zp
 λ(Hξ (η))

called the p-adic height pairing affiliated with the Zp -extension. There is aunique extension of this pairing to the group R(E/K):
R(E/K) × R(E/K)

 Qp

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CHRISTIAN WUTHRICH

The p-adic regulator of the fine Selmer group Reg λ (R(E/K)) is defined tobe the determinant of this pairing, which is well-defined up to multiplication
by a unit in Zp . The choice of the surjective morphism λ only changes theregulator by a unit. The pairing is non-degenerate if and only if the regulator
is non-zero.
For the rest of the article the following conjecture due to Perrin-Riou,see [11, Conjecture 3.3.7.B.i], will play a crucial role.
Conjecture 5.2. The p-adic regulator Reg cyc (R(E/K)) of the fine Selmer
group affiliated with the cyclotomic Zp -extension is non-zero.
Starting with the same element ξ, we could also construct an extension0 Qp/Zp (1) Wξ

 E{p}

0

defined as before using the Weil pairing E{p} ⊗ E{p}
 Qp/Zp (1). Thanks
to the assumption on the class group, there is a diagram like before HΣ1 K, Qp/Zp (1) HΣ1 (K, Wξ )
 HΣ1 (K, E{p})0g
0  ⊕Σ H Kυ , Qp/Zp (1)1g ⊕Σ H (Kυ , Wξ )1g
 ⊕Σ H (Kυ , E{p})1

0and the snake lemma yields a map
Hξ : R(E/K)

 HΣ1 (K, Zp )∧ .

The Zp -extension given by λ ∈ Hom(GΣ (K), Zp ) = HΣ1 (K, Zp ) allows us todefine a second pairingh·, ·iλ : RΣ (E/K) × R(E/K)(ξ, η)

 Qp/Zp
 Hξ (η)(λ)

It is immediate that the dualâ : Hom(R(E/K), Qp/Zp ) HomZp (RΣ (E/K), Zp )of the map a defined in (3.1) sends the second pairing Hξ to the first pairingHξ for all ξ in RΣ (E/K).
5.2. Iwasawa theoretic height. The previously defined pairing on thefine Selmer group can be decomposed into a sequence of maps that will link
the regulator to the Euler-characteristic of the fine Selmer group.
Fix a topological generator γ of the Galois group Γ of our Zp -extension
1
given by λ, i.e. λ(γ) ∈ Z×
p . With this choice, we may identify H (Γ, M ) withthe coinvariance MΓ for all Γ-modules M .
The restriction provides us with a map b : R(E/K)
 R(E/∞ K)Γ as in
Proposition 4.1. The identity induces a map
c : R(E/∞ K)Γ

 R(E/∞ K)Γ

IWASAWA THEORY OF THE FINE SELMER GROUP

11

and there is a map
d : R(E/∞ K)Γ

 HΣ1 (∞ K, E{p})Γ

coming from the natural inclusion. The spectral sequence of Hochschild-Serregives a transgression map(5.2)

e : HΣ1 (∞ K, E{p})Γ   HΣ2 (K, E{p})whose cokernel is equal to HΣ (∞ K, E{p})Γ . The global duality in section 2.1
states that there is an isomorphism(5.3)

f : HΣ2 (K, E{p})∧∼= RΣ (E/K).

Proposition 5.3. The p-adic height pairing on the fine Selmer grouphλ : RΣ (E/K)

 HomZp (RΣ (E/K), Zp )corresponding to the Zp -extension λ, given by hλ (ξ)(η) = hξ, ηiλ is equal tothe composition in the commutative diagram
HomZp (RΣ (E/K), Zp )∧
(5.4)

ĥλ

 RΣ (E/K)∧ffag
R(E/K)bg
R(E/∞ K)ΓHΣ2 (K, E{p})fec R(E/∞ K)Γ

d HΣ1 (∞ K, E{p})Γ

The proof of this proposition is given in section 4.4 of [10] or, in our notations, in [17, I.6].
5.3. The analytic height pairing. In [2] an analytic version of the padic height pairing on the fine Selmer group is used to compute some cases ofthe main conjecture for curves with supersingular reduction. We give here aslightly different formula.
We may suppose the elliptic curve E is given in a Weierstrass equation
with integral coefficients. According to Proposition 2 in [18], there exists
a subgroup of finite index in E(K), denoted by E • (K), of points P whose
coordinates can be represented as fractions of coprime integers, i.e.

a(P ) b(P ),P =e(P )2 e(P )3for some elements a(P ), b(P ) and e(P ) in the ring of integers OK of K, defined
up to multiplication by a unit in O×K.

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CHRISTIAN WUTHRICH

Proposition 5.4. Let P and Q be two elements in R(E/K) which arein the image of the Kummer map κ : E(K) ⊗ Zp
 HΣ1 (K, Tp E). Let{Pk }k and {Qk }k be sequences of points converging to P and Q in E(K) ⊗ Zp
respectively. Assume that Pk and Qk belong to E • (K) for all k. Then thecyclotomic p-adic height pairing of P and Q is equal toa(Pk ) · a(Qk )1· lim log ◦NK:Q
hP, Qicyc =2p k→∞ pa(Pk + Qk )where logp denotes the p-adic logarithm.
For a proof of this formula we refer the reader to [17, Theorem IV.8]. If thep-primary part of the Tate-Shafarevich group X(E/K) is finite, then there isa subgroup of finite index in R(E/K) for which the above formula applies andhence one could use this formula for computing the p-adic regulator. Nevertheless there is a faster algorithm for its computation using the σ-function of
Bernardi [1] and the p-adic elliptic logarithm. For details see section VI.1.1
in [17].
Note also that this formula proves the fact that the pairing is indeed bilinear
and symmetric.

6. The Euler-characteristic
Using the choice of a topological generator γ of Γ, we identify Λ with Zp [[T ]]by sending γ − 1 to T . A torsion Λ-module X gives rise to a characteristic
power series fX which is a non-zero element of Λ, defined up to a unit in Λ× .
The leading coefficient of fX at T = 0,
fX∗ (0) =fX (T )
∈ Zp \ {0}T ordT (fX ) T =0

is called the Euler-characteristic of the Λ-module X. Its valuation is independent of the choice of γ.
Theorem 6.1. Let E be an elliptic curve defined over a number field
K. Suppose the p-adic height affiliated with a Zp -extension ∞ K : K is nondegenerate. Then
(1) The dual of R(E/∞ K) is a torsion Λ-module.
(2) Let r be the corank of R(E/K). Then the characteristic power seriesfR (T ) of the dual of R(E/∞ K) has a zero of order r at T = 0.
(3) There is an injectionΓ
R(E/K)   ∞ H 2 (E/K)Γ = lim HΣ2 (n K, Tp E)←−
whose cokernel J is finite.

IWASAWA THEORY OF THE FINE SELMER GROUP

13(4) The leading coefficient of fR has the same valuation as#Tloc · # R(E/K)/ div∗(mod Z×fR (0) ≡ Regλ (R(E/K)) ·p)#Tgl · #J · #I
where I is the index of R(E/K) in Tp R(E/K) defined in section 3
and Tgl and Tloc were defined in (4.2).
The formula can be simplified substantially if one assume more restrictive
hypotheses. The following corollaries specify the theorem to cases when the
Zp -extension is cyclotomic. The first three conclusions of the theorem arestill valid and so only a reformulation of the leading coefficient of fR is given.
But first, we need some more definitions: the subgroup of elements in R(E/K)
 HΣ1 (K, E{p})that belong to the image of the Kummer map E(K)⊗ Qp/Zp
will be denoted by M(E/K). The fine Tate-Shafarevich group (E/K) is
defined to be the quotient of R(E/K) by M(E/K). It naturally identifies witha subgroup of the p-primary part of the Tate-Shafarevich group X(E/K). For
properties of this group we refer the reader to [19].
Corollary 6.2. Assume that E has potentially good reduction at all place
above p and that the fine Tate-Shafarevich group (E/K) is finite. If thecyclotomic p-adic height is non-degenerate then the leading coefficient of thecharacteristic power series fR of the dual of the fine Selmer group R(E/∞ K)
is equal toQ# TorsZp (D) · υ-p cυ · #(E/K)∗(mod Z×fR (0) ≡ Regcyc (R(E/K)) ·p)#Jwith D being the cokernel of the localisation map from E(K)⊗Zp to the p-adic
completion of ⊕υ|p E(Kυ ).
Proof. By Lemma 4.2 we know that Tgl has the same order as E(K){p}and Lemma 4.3 tells us thatY
YYY
#Tloc =c(p)#E(Kυ ){p} ≡cυ ·#E(Kυ ){p} (mod Z×
υ ·
p)
υ|p

υ-p

υ-p

υ|p

(p)

under our assumptions. Here cυ is the highest power of p dividing cυ . If
the fine Tate-Shafarevich group is finite, then R(E/K) coincides with its
subgroup M(E/K) defined to be the elements in the image of the Kummer HΣ1 (K, Tp E). Similarly Tp R(E/K) can be replaced by
map E(K) ⊗ Zp
Tp M(E/K). Now the Theorem 7.1 in [19] can be used to compute I: it statesthat there is an exact sequence
(6.1)

0

I

T

 TorsZp (D) M(E/K)/ div

0

14

CHRISTIAN WUTHRICH

with T beings the quotient of ⊕υ|p E(Kυ ){p} by the global torsion points. By
the finiteness of (E/K) we have a short exact sequence
(6.2)

0

 M(E/K)/ div

 R(E/K)/ div (E/K)

0and hence we can compute modulo Z×p#(M(E/K)/ div) · #(E/K)#Tloc #(R(E/K)/ div) Ycυ · #T ·
·
≡
#Tgl
#I
#I
υ-p

≡

Y

cυ · #I · # TorsZp (D) ·υ-p#(E/K)#I

which finished the proof of the corollary.

Finally we specify to the case of a curve over Q and treat the cases of rank 0,
1 and strictly bigger than 1 separately.
Corollary 6.3. Let E be an elliptic curve defined over Q with potentiallygood reduction at p whose Tate-Shafarevich group X(E/Q) is finite. The fineSelmer group R(E/Q) is trivial if the rank of E(Q) is less than 2 and its rankis equal to rank(E(Q)) − 1 otherwise. If E(Q) is finite thenQ
#E(Qp ){p} · cυ · #(E/Q)(mod Z×fR (0) ≡p ).
#E(Q){p} · #J
If the rank of E(Q) is equal to 1 thenQ
#D · cυ · #X(E/Q){p}fR (0) ≡#J(mod Z×p ).

If E(Q) has rank strictly larger than 1 and the p-adic height is non-degenerateon the fine Selmer group thenQ
#D · cυ · #X(E/Q){p}(mod Z×fR∗ (0) ≡ Regcyc (R(E/Q)) ·p ).
#J
Proof. These formulae are simply obtained by specifying the formula in
the previous corollary. If the rank of E(Q) is smaller than 2, then R(E/Q) is
trivial and hence the p-adic height pairing is automatically non-degenerate ofregulator equal to 1. If the rank is zero then D is the quotient of E(Qp ) by
E(Q){p}. If the rank is positive, then D has to be torsion and by Theorem 3.5
in [19] the fine Tate-Shafarevich group (E/Q) coincides with the p-primarypart of X(E/Q). Finally the assumption that the reduction is potentiallyQ
good and that p 6= 2 assures that cp is not divisible by p and the term υ-p cυ
has the same valuation as the product of all Tamagawa numbers.
IWASAWA THEORY OF THE FINE SELMER GROUP

15

7. Proof of Theorem 6.1
The first two assertions are well-known. The first conclusion, that thedual Y of the fine Selmer group R(E/∞ K) is Λ-torsion is usually called theweak Leopoldt conjecture; it is equivalent to the vanishing of HΣ2 (∞ K, E{p})(see Proposition 1.3.2 in [12]). The two conclusions are now contained in
Corollaire 3.4.3 of [12]. A different proof is presented in I.7 and I.8 of [17].
In particular, we deduce that the map e in (5.2) whose cokernel is equalto HΣ2 (∞ K, E{p})Γ must be an isomorphism. By Proposition 4.1, the map b
has finite kernel and cokernel, by (3.2) and (5.3) the same is true for a andf . Since hλ has the same property by the assumption on the non-degeneracyof the height, all maps in the diagram (5.4), have finite kernel and cokernel
except maybe c and d. It follows that the kernel of c is finite. Hence themap ĉ : Y Γ YΓ has finite cokernel and this, together with the fact that
Y is Λ-torsion, is sufficient to show that ĉ has also finite kernel and that theleading coefficient of fR has the same valuation as z(c). In fact it can be
deduced from the decomposition theorem for finitely generated Λ-modules.
(See Lemma I.28 in [17] or Lemme 0.2.3 in [10]). It follows now that d has
finite kernel and cokernel, too.
Note that the target of d is isomorphic (via e and f ) to RΣ (E/K)∧ which
is known to be a divisible group by Lemma 3.1. Hence the finite cokernel ofd is trivial. We need the following
Lemma 7.1. There is a surjective map⊕Σ H 1 (∞ Kυ , E{p})Γ

g

 (∞ H 1 (E/K)∧ )Γ .

Proof. Define the group ∞ C by the exact sequence
(7.1)

0

 R(E/∞ K) HΣ1 (∞ K, E{p})

 ∞C

0

The global duality in section 2.1 over ∞ K allows us to complete this by anotherexact sequence
(7.2)

0

 ∞C

 ⊕Σ H 1 (∞ Kυ , E{p}) ∞ H 1 (E/K)∧

0where the last zero comes from the proven weak Leopoldt conjecture, becauseHΣ2 (∞ K, E{p}) is trivial. So from (7.1) we conclude that
0
(7.3)

 R(E/∞ K)Γ
 R(E/∞ K)Γ

d HΣ1 (∞ K, E{p})Γ

 ∞CΓ



 HΣ1 (∞ K, E{p})Γ

 ∞ CΓ

0is an exact sequence in which the surjective map d appears. Hence ∞ CΓ istrivial. From (7.2) we see thatΓ(7.4) 0 ∞C Γ ⊕Σ H 1 (∞ Kυ , E{p})Γ
 ∞ H 1 (E/K)∧
0

16

CHRISTIAN WUTHRICH

and the lemma follows.
We will need the short exact sequence(7.5)

 ∞ H 1 (E/K)Γ

0cor

 HΣ1 (K, Tp E) ∞ H 2 (E/K)Γ

0

which is a consequence of the spectral sequence of Tate, see Theorem II.1.11
and II.1.12 in [9]. The dual of this sequence can now be put together withthe two exact sequences (4.2) in a larger commutative diagram:
0
f

0
f

0
f

HΣ1 (∞ K, E{p})Γ ◦f ⊕Σ H 1 (∞ Kυ , E{p})Γ ◦f (∞ H 1 (E/K)Γ )∧f

HΣ1 (K, E{p}) ◦f ⊕Σ H 1 (Kυ , E{p}) ◦f HΣ1 (K, Tp E)∧
f

Tgl ◦
f

 Tloc ◦f (∞ H 2 (E/K)Γ )∧
f

0

0

0

Here the vertical lines are exact sequences; the horizontal lines are not exact
but they form at least a cochain complex (completed with zeros to the left
and the right), which is symbolised with the arrows of the form ◦  . Notethat the top line is the Γ-invariance of a part of the global duality over ∞ K
as we used it before for the definition on ∞ C. In order to recover the bottom
complex, we take the second sequence exposed in 2.1(7.6)
 E(∞ K){p} ∞T ∞ H 2 (E/K)∧ R(E/∞ K)00where ∞ T is ⊕Σ H 0 (∞ Kυ , E{p}). The Γ-coinvariance of this sequence gives
the desired complex(7.7) 0

 Tgl ◦  Tloc ◦  (∞ H 2 (E/K)Γ )∧ R(E/∞ K)Γ

 0.

The middle complex of the above huge diagram is actually simply a partof the global duality sequence over K and, hence, is exact in the middle.
The long exact sequence associated to this short exact sequence of cochain
complexes breaks up into two exact sequences with four terms. Denote the•••cochain complexes by Ztop
, Zmi
and Zborespectively and fix notations so that
their zero-th term is on the middle vertical line. Comparing with (4.1), weIWASAWA THEORY OF THE FINE SELMER GROUP

17

get that the first part is simply•
 H −1 (Zbo
)
w
w
w
(7.8)
w

0

0

 ker(b)• H −1 (Zmi
)
w
w
w
w

 R(E/K)b•
 H −1 (Ztop
)
w
w
w
w

 R(E/∞ K)Γ

• H 0 (Zbo
)
w
w
w
w

 coker(b)

0

0

•
Lemma 7.1 shows that H 1 (Ztop) = 0. The sequence (7.4) proves that ∞ C Γ is1•0
 Ztop ; together with (7.3) this yields H 0 (Ztop) = ker d.
the kernel of Ztop
1•
Finally the sequence (7.7) shows that H (Zbo ) is equal to R(E/∞ K)Γ , hence
we have00•
 H 0 (Ztop
)
w
w
w
w

 ker(d)• H 1 (Zbo
)
w
w
w
w
w

 R(E/∞ K)Γ

• H 1 (Zmi
)
w
w
w
w

 RΣ (E/K)∧•
 H 1 (Ztop
)
w
w
w
w
w

0

0

We shall prove now the third statement of Theorem 6.1. Since ker(d) is
finite, the previous exact sequence shows that there is an injection of RΣ (E/K)
into the dual of R(E/∞ K)Γ with finite cokernel. (It is actually not difficult toˆ ê◦ f.)
ˆ Then from the sequence (7.7),
show that this map is the composition d◦
one deduces an injection of the dual of R(E/∞ K)Γ into ∞ H 2 (E/K)Γ whose
cokernel lies in the finite group Tloc , see Lemma 4.3. Thus we have an injectionof RΣ (E/K) into the group ∞ H 2 (E/K)Γ with image of finite index.
There is also a map from R(E/K) into ∞ H 2 (E/K)Γ obtained by sending
R(E/K) into HΣ1 (K, Tp E), followed by the second map in the sequence (7.5)
coming from Tate’s spectral sequence. This map restricts to the previous
injection on RΣ (E/K). Since R(E/K) is Zp -free by Lemma 3.1, the kernel
 ∞ H 2 (E/K)Γ must be trivial.
of R(E/K)
Finally, we come to the last part of the theorem, namely the computationof fR∗ (0) = z(c). From the previous computations, we may compute the orderof J as#J · [R(E/K) : RΣ (E/K)] = [∞ H 2 (E/K)Γ : RΣ (E/K)]= # ker(d) · [(∞ H 2 (E/K)Γ )∧ : R(E/∞ K)Γ ]= # ker(d) ·•#Tloc · #H 1 (Zbo).
•
0
#Tgl · #H (Zbo )18

CHRISTIAN WUTHRICH

In the last line, we have used the complex (7.7). The identification in (7.8)
allows us to rewrite the expression as#Tloc · # ker(b)#Tgl · # coker(b)#Tloc= z(d) · z(b) ·#Tgl#J · [R(E/K) : RΣ (E/K)] = # ker(d) ·(7.9)

We now compute z(c) using the decomposition of the p-adic height in (5.4),(7.9), (3.2) and the fact that e and f are isomorphisms.
z(c) =z(ĥ)
z(f ) · z(e) · z(d) · z(b) · z(a)#Tloc·
#Tgl · #J · [R(E/K) : RΣ (E/K)]#(R(E/K)/ div)·
[R(E/K) : RΣ (E/K)] · #I

= Regλ (RΣ (E/K)) ·If the regulator on RΣ (E/K) is replaced by Regλ (R(E/K)) times the squareof the index [R(E/K) : RΣ (E/K)], the index cancels from the formula. We
obtain the formula in the theorem, which is now independent of the chosenset Σ.


8. On the µ-invariant
Proposition 8.1. Let E be an elliptic curve over Q which admits an
isogeny defined over Q of degree p. Then the µ-invariant of the dual ofR(E/∞ Q) is zero.
Proof. Let C the kernel of the isogeny. The fixed field K of the kernel of the Aut(C) is a cyclic extension of Q over which E admits
map Gal(Q̄ : Q)a p-torsion point. By Corollary 3.6 of [5] this is enough to guarantee thatthe µ-invariant over K is trivial. (The proof relies on the theorem of FerreroWashington on the triviality of the classical µ-invariant.) The µ-invariant overQ can only be smaller.

Coates and Sujatha made the following
Conjecture 8.2. Let E/K be an elliptic curve over a number field. Thenthe µ-invariant of the fine Selmer group with respect to the cyclotomic Z p extension is zero.
We will see in the examples that our computation provide ample numericalevidence for this conjecture over Q.

IWASAWA THEORY OF THE FINE SELMER GROUP

19

9. Curves of rank 0
In this and the following two sections, we wish to give several numericalexamples in order to explain what one can obtain from the Euler-characteristic
formula in Corollary 6.3. The curve E will always be defined over Q and wewill frequently assume that the Tate-Shafarevich group is finite.
The first part is concerned with elliptic curves E of rank 0 over Q. Since
M(E/Q), contained in E(Q) ⊗ Qp/Zp , is trivial, we see that R(E/Q) is equalto (E/Q). On the compact side, we know that R(E/Q) is zero if and only(E/Q) is finite; in which case, the group J is therefore nothing else but

J = coker cor: ∞ H 1 (E/Q)Γ

 HΣ1 (Q, Tp E) = ∞ H 2 (E/Q)Γ .

Proposition 9.1. Suppose X(E/Q) is finite. If E(Q) is finite but nontrivial, then R(E/∞ Q) and J are trivial for all primes p > 7 not dividingQυ cυ or #X(E/Q).
Proof. For p > 5, we have 0 < #Ẽns (Fp ) < 2p by the theorem of HasseWeil. Since there is a point of finite order 1 < ` 6 7 defined over Q, thenumber of points in the reduction must be divisible by ` and hence can not beQequal to p unless ` = p. Therefore for a prime p > 7 dividing cυ , there can
not be a point of order p on E(Qp ). Hence the numerator in the first formula ofCorollary 6.3 contains only factors which are not divisible by p. This formula
only applies if E has potentially good reduction at p; but if not the numerator
is still a bound on the product #J · fR (0). Hence #J = fR (0) = 1, and sofR ∈ Λ× implies that R(E/Q) is finite.
Moreover we see from Proposition 4.1, we see that b is an isomorphism,i.e. R(E/Q)Γ is trivial. This implies that R(E/∞ Q) is trivial because it isfinite.

If there is no torsion point defined over Q, then it is still true that the fineSelmer group over ∞ Q is most often trivial:
Proposition 9.2. For an elliptic curve E/Q of rank 0 with finite X(E/Q),
there is a set of density of 1 of primes for which R(E/∞ Q) is finite. If E
admits an isogeny of degree p to a curve with a non-trivial torsion point, we
may take the set to include all but a finite number of primes.
Proof. The proof of Proposition 5.1 in Greenberg’s part of [3] can be applied
to the formula in Corollary 6.3.


20

CHRISTIAN WUTHRICH

9.1. Numerical examples. It is quite traditional to consider as a firstexample the curves of conductor 11. They are given by the three equations
E1 :y 2 + y = x3 − x2 − 10 x − 20

E2 :y 2 + y = x3 − x2 − 7820 x − 263580

E3 :y 2 + y = x 3 − x2

For all good, ordinary primes p 6= 5, the Selmer group S(Ei /∞ Q) is trivial(see [4, Theorem 4.6]), and for all supersingular primes it is isomorphic to Λ(see [4, Theorem 4.5]). For p = 5, we haveS(E1 /∞ Q)∧ = Λ/(5)S(E2 /∞ Q)∧ = Λ/(52 )S(E3 /∞ Q)∧ = 0

Proposition 9.3. The fine Selmer groups R(Ei /∞ Q) of the curves of conductor 11 are trivial for all odd primes p and all i, except for E 1 and p = 5
in which case it is finite but non-trivial and J contains 5 elements.
Proof. It is known that the Tate-Shafarevich groups of the three curves Ei
are trivial, see [17, Lemma VI.3]. Let p be a prime not dividing 2 · 5 · 11 and,hence, of good reduction. The first formula in Corollary 6.3 gives#J · fR (0) ≡ #Ei (Qp ){p} (mod Z×p)because the product of the Tamagawa factors is equal to 5 (for E1 only) or 1
and there are no global p-torsion points for p 6= 5. On E1 there is a rational
5-torsion point. This implies just as in the proof of Proposition 9.1 that p > 7
is non-anomalous, hence we can check that Ei (Qp ){p} is trivial for all p 6= 5.
So for all these primes J and R(Ei /∞ Q) are trivial.
For p = 11, one can deduce that #J · fR (0) = 1 in the same manner, butusing the formula in Theorem 6.1 which hold also for primes of bad reduction.
Finally for p = 5, we know that the characteristic power series of the Selmer
groups are powers of 5. But 5 can not divide the series fR by Proposition 8.1
because of the presence of isogenies of degree 5. Hence we have also here that
R(Ei /∞ Q) is finite. On the other hand, we compute that #J · fR (0) = #J isequal to 5 for E1 and 1 for the other curves. For i = 2 and 3, Proposition 4.1
proves that R(Ei /∞ Q)Γ is trivial, whence so is R(Ei /∞ Q). Meanwhile for
i = 1 we have #R(E1 /∞ Q)Γ = 5.
Another more complicated example is the curve 182D1 given byE:

y 2 + x y + y = x 3 − x2 + 3 x − 5and the prime p = 5. Unlike in the previous case we do not have an isogenyof degree p at our disposition. The Mordell-Weil group E(Q) is trivial andQ
the product cυ is equal to 1. The 5-primary part of the Tate-Shafarevichgroup X(E/Q) can be verified to be trivial using the Heegner point over

IWASAWA THEORY OF THE FINE SELMER GROUP

21

the field Q(i). Although there is no 5-torsion point in E(Q), there is one inE(Q5 ), hence we have #J · fR (0) = 5. Even though we can not conclude
whether R(E/∞ Q) is finite or not, we can still decide that Conjecture 8.2
holds. Since E[p] is an irreducible Galois-representation, the theorem of Kato
shows that fS divides the p-adic L-function. The invariants of this latter havebeen computed by Pollack [13] and he claims that µS = 0. Since fR divides
fS , we must have that µR = 0. The λ-invariant of the usual Selmer groupequals at most 2.

10. Curves of rank 1
Compared to the computations for curves of rank 0, the only big differenceis the presence of the term D in the formula in Corollary 6.3. Let E be thecurve 37A1
E : y 2 + y = x3 − x.
It is known that E(Q) is a free group generated by P = (0, 0) and X(E/Q) =0, see [17, Lemma VI.5.]. We choose the good ordinary non-anomalous prime
p = 179 which is the smallest prime for which D is non-trivial, see (6) in [19].
The formula gives now #J · fR (0) = p. But since the prime p has goodordinary reduction, we may use the Euler-characteristic formula (1.1) for theusual Selmer group and we get that fS∗ (0) is a unit and hence fS = T . Hence we
can deduce that fR = 1 and that J has p elements. In particular R(E/∞ Q)
is finite but non-trivial. The same way one proves the following, see [17,Proposition VI.6.].
Proposition 10.1. The fine Selmer group R(E/∞ Q) for the curve 37A1
is finite for all odd primes p < 1000.
We do not know anything about the distribution of primes p for a fixed
elliptic curve E for which the group D is non-trivial. But it seems that for
curves of rank 1, there might be an infinite set of such p but probably ofdensity 0. Nevertheless even for these primes, the above method consisting ofcomparing with the usual Selmer group gives for most of them that the fineSelmer group R(E/∞ Q) is finite. This is in support of the Conjecture 1.2.
More computations can be found in [17, Table VI.2].
Here is a supersingular example:E:y 2 + x y + y = x 3 − x2 .

This is the curve labelled 53A1 whose Mordell-Weil group is generated by P =(0, 0). Conjecturally the Tate-Shafarevich group is trivial; it can be shownwith a 3-descent that X(E/Q)[3] is trivial. We consider the supersingular

22

CHRISTIAN WUTHRICH

prime p = 3. There are no 3-torsion points on E(Q3 ), but D is of order
3, because the first multiple of P in the kernel of reduction at 3 is 7 · P =40, − 1025( 81729 ) which is already in the second layer of the formal group. Hence#J · fR (0) = 3. Now, Pollack [13] computes for this example the Iwasawa
invariants of the 3-adic analytic L-functions of the + and − Selmer groups as
defined in [14]. They are both equal to T . Using Kobayashi’s result [8] on themain conjecture in the supersingular context and that fR has to divide bothof these analytic L-function, we draw the conclusion that fR is trivial and soR(E/∞ Q) is, once again, finite.

11. Curves of rank > 2
The situation is similar to the rank 1 case. One more factor is appearing,
due to the fact that the fine Selmer group R(E/Q) is no longer finite, namely
the regulator. The non-degeneracy of the p-adic height implies that R(E/∞ Q)is Λ-torsion. But over Q this is known unconditionally by the much celebrated
theorem of Kato. We still need to compute the p-adic height for computing abound on the leading coefficient fR∗ (0).
The methods for proving that the characteristic power series fR is equal to
T r where r is the corank of R(E/Q) are the same as before : If the formulafor #J · fR∗ (0) does not prove it, we will try to compare it with the classicalformula (1.1) for fS∗ (0). The last possibilities would be to use either an isogenyof degree p, if there is one defined over Q, or to use analytic computations by
Pollack [13] and the divisibility of Kato. But all these methods are only usefulto give upper bounds on the Iwasawa-invariants of R(E/∞ Q). For tables anddetails of the computation, we refer the reader to [17, Table VI.3].
So far, we did not present any example in which the characteristic power
series fR is not simply a power of T . The last example gives now a curve witha different behaviour. In this case one can prove that the fine Selmer group
has to grow because the rank of the fine Mordell-Weil group increases. Thereis no known example yet of an elliptic curve E/Q and a prime p for which thegroup (E/∞ Q) is infinite. Compare this with the Question 8.3 in [19].
The following example is a rather complicated case for which the characteristic series of the fine Selmer group is not trivial. Let E be the curve 5692A1
given byE:

y 2 = x3 + x2 − 18 x + 25.

The Mordell-Weil group is a free group generated by the points P1 = (0, 5)
and P2 = (1, 3). We are interested in this curve for the prime p = 3. It has

IWASAWA THEORY OF THE FINE SELMER GROUP

23

good ordinary, anomalous reduction at p, but E(Q3 ){3} = 0. The analytic
order of X(E/Q) is 1 and the product of Tamagawa numbers is 3.
Assuming the triviality of the 3-primary part of the X(E/Q), we get thatthe fine regulator has valuation 5 and the formula in Corollary 6.3 shows that
#J · fR∗ (0) has valuation 6. Since it has ordinary reduction, we may comparewith the characteristic series of the Selmer group. The first coefficient fS∗ (0)is 33 by (1.1) if X(E/Q) has no 3-torsion.
Proposition 11.1. Assume that the 3-primary part of X(E/Q) is trivial.
Then the characteristic power series of the Selmer group is equal tofS = T 2 · (3 + 3T + T 2 )2 · (3 + 9T + 18T 2 + 21T 3 + 15T 4 + 6T 5 + T 6 )= ((1 + T )3 − 1) · ((1 + T )9 − 1)
The characteristic power series for the dual of the fine Selmer group isfR = T · (3 + 3T + T 2 ) = (1 + T )3 − 1.
The group J contains 35 elements. In particular, we have that E(∞ Q) has
rank 12, M(E/∞ Q) has rank 3 and X(E/∞ Q) is finite.
Proof. Let 1 Q be the first layer of the cyclotomic Z3 -extension of Q; it isgenerated by α satisfying α3 − 3α + 1 = 0. With some luck, we were able tofind six independent points in E(1 Q).
P3 =(−α2 − 2 · α + 2, − 3 · α2 − 3 · α + 4),P4 =(−α2 − 2 · α + 3, − 2 · α2 − 2 · α),P5 = (−2 · α2 − 3 · α + 4, − α2 + 2),
P6 = (−2 · α2 − 2 · α + 6, − 2 · α2 + 2 · α + 9).
Hence the rank of E(1 Q) is at least 6. Next, we are looking at the second
layer 2 Q defined by β with β 3 − 3 β = α. Again we are lucky and find a point
P7 = ( − β 6 + 7 · β 4 − β 3 − 14 · β 2 + 4 · β + 7,
β 7 − 3 · β 6 − 7 · β 5 + 20 · β 4 + 11 · β 3 − 36 · β 2 + β + 15)
which is linearly independent from the previous six points. The rank of E(2 Q)
has to be at least 12.
The series fS is therefore divisible by the right hand side of the formula in
the proposition. Since fS∗ (0) = 33 , this divisibility must be an equality.
Because the rank of the Mordell-Weil group is jumping from rank E(Q) = 2
to rank E(1 Q) = 6, it is clear that the fine Mordell-Weil group increases its
rank from 1 to 3, in other words (3 + 3T + T 2 ) divides exactly once fR . It is
not possible that the last factor divides fR .


24

CHRISTIAN WUTHRICH

The analytic 3-adic L-function can be shown to be divisible by the three
irreducible factors in the formula, but by no other distinguished polynomial
or by p. I am thankful to Robert Pollack for this computation.

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, The fine Tate-Shafarevich group, to appear in Math. Proc. Cambridge Philos.
Soc., 2005.

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Séction de mathématiques, CSAG, École polytechnique fédérale, 1015 Lausanne, Switzerland
E-mail address: christian.wuthrich@epfl.ch